Theorem bitr3VD 42469

Description: Virtual deduction proof of bitr3 353. The following user's proof is completed by invoking mmj2's unify command and using mmj2's StepSelector to pick all remaining steps of the Metamath proof.

1::	⊢ (   (𝜑 ↔ 𝜓)   ▶   (𝜑 ↔ 𝜓)   )
2:1,?: e1a 42247	⊢ (   (𝜑 ↔ 𝜓)   ▶   (𝜓 ↔ 𝜑)   )
3::	⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜑 ↔ 𝜒)    ▶   (𝜑 ↔ 𝜒)   )
4:3,?: e2 42251	⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜑 ↔ 𝜒)    ▶   (𝜒 ↔ 𝜑)   )
5:2,4,?: e12 42344	⊢ (   (𝜑 ↔ 𝜓)   ,   (𝜑 ↔ 𝜒)    ▶   (𝜓 ↔ 𝜒)   )
6:5:	⊢ (   (𝜑 ↔ 𝜓)   ▶   ((𝜑 ↔ 𝜒) → (𝜓 ↔ 𝜒))   )
qed:6:	⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ↔ 𝜒) → (𝜓 ↔ 𝜒)))

(Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
bitr3VD	⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ↔ 𝜒) → (𝜓 ↔ 𝜒)))

Proof of Theorem bitr3VD

Step	Hyp	Ref	Expression
1		id 22	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 ↔ 𝜓))
2	1	bicomd 222	. 2 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 ↔ 𝜑))
3		id 22	. . 3 ⊢ ((𝜑 ↔ 𝜒) → (𝜑 ↔ 𝜒))
4	3	bicomd 222	. 2 ⊢ ((𝜑 ↔ 𝜒) → (𝜒 ↔ 𝜑))
5		biantr 803	. . 3 ⊢ (((𝜓 ↔ 𝜑) ∧ (𝜒 ↔ 𝜑)) → (𝜓 ↔ 𝜒))
6	5	ex 413	. 2 ⊢ ((𝜓 ↔ 𝜑) → ((𝜒 ↔ 𝜑) → (𝜓 ↔ 𝜒)))
7	2, 4, 6	syl2im 40	1 ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 ↔ 𝜒) → (𝜓 ↔ 𝜒)))

Syntax hints:   → wi 4   ↔ wb 205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-an 397

This theorem is referenced by: (None)